A common question I get in my divorce practice is, can I change my child's name to my new husband's name. The answer to that question is yes. However, in order to do so you will need the consent of the biological father.
What if you do not know where the father is?
They are still entitled to notice. You must exert all efforts to locate them (web search, calling friends and family, etc...) If you cannot find them then you can provide them notice in a newspaper. You will have to check with your local court for the costs and how to do this.
What if I know where the father is and he will not agree?
Then you probably will be unsuccessful in changing the child's name. Absent an adoption of the child, the courts will typically leave the child's name the same as that of their father.
If you have a situation that I have not discussed, contact a local lawyer for help.